Scaling behavior of chaotic systems with riddled basins.

Recently it has been shown [1,2] that chaotic systems with a simple, very common type of symmetry can display a striking new kind of behavior. In particular, these systems may have an attractor whose basin of attraction is such that every point in the basin has pieces of another attractor s basin arbitrarily nearby [3]. That is, if rp is any point in the first attractor's basin, then the phase space ball of radius e centered at ro has a nonzero frac-. tion of its volume lying in a another attractor's basin, and this is so no matter how small e is. Thus there is always a positive probability that an arbitrarily small uncertainty in ro will put the initial condition in another attractor s basin [4]. We say that the first basin is riddled by the second basin. This paper reports quantitative theoretical results [5] for riddled basins and compares these results with numerical experiments on a simple mechanical system [2]. The theoretical results apply near the critical point where a riddled basin G.rst appears as a system parameter is varied. In this parameter regime the behavior becomes universal [6] in the sense that it is controlled by a few gross system variables. We emphasize, however, that the qualitative behavior found persists away from the transition. A general set of conditions under which riddled basins can occur are as follows: (i) There is an invariant subspace M whose dimension dM is less than that of the phase space d~, . (ii) The dynamics on the invariant subspace M has a chaotic attractor A for initial conditions on M. (iii) For typical orbits on A the Lyapunov exponents for infinitesimal perturbations in the directions transverse to M are negative, so that A is also an attractor in the full d» dimensional phase space. (iv) At least one of the transverse I yapunov exponents, although negative for almost any orbit on A, experiences G.nite time fiuctuations that are positive. (v) There is another attractor not in M. In order to illustrate the above, we refer to an example. In particular, we consider the motion of a point particle of unit mass moving in a two-dimensional potential V(z, y) subject to dynamic friction (coefficient v) and sinusoidal forcing,